Bimaterial Interface Cracks Research Articles

Stress intensity factor analyses of three-dimensional interface cracks using tetrahedral finite elements

A numerical method is proposed for evaluating the stress intensity factors of a three-dimensional bimaterial interfacial crack using tetrahedral finite elements. This technique is based on the M1-integral method and employs the moving least-squares approximation. Stress or strain in the M1-integral equation is automatically approximated from the nodal displacements obtained by the finite element analysis using the moving least-squares method. Therefore, the presented method needs no elemental information from the finite element analysis. In this study, stress intensity factor analyses of some typical three-dimensional interface crack problems using the tetrahedral finite elements are demonstrated.

A novel singular finite element on mixed-mode bimaterial interfacial cracks with arbitrary crack surface tractions

This paper investigates interfacial cracks with arbitrary crack surface tractions. A novel singular finite element which is constructed with the analytical solution around interfacial cracks is presented. Interfacial crack problems can be analyzed numerically using the singular finite element, and Mode I and/or Mode II stress intensity factors can be obtained directly. Unlike other enriched elements for cracks, neither extra unknowns nor transition elements are required. Numerical examples are given to illustrate the validity of present method.

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.

Stress Intensity Factor for Interfacial Cracks in Bi-Materials Using Incompatible Numerical Manifold Method

Incompatible numerical manifold method (INMM) uses interpolation functions based on the concept of partition of unity, and considers the asymptotic solution and the discontinuity of displacement. This paper describes the application of INMM to bi-material interfacial crack. The two dimensional near-tip asymptotic displacement functions are added to the trial function approximation. This enables the domain to be modeled by manifold elements without explicitly meshing the crack surfaces. The crack-tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The INMM facilitates the incorporation of the oscillatory nature of the singularity within a conforming manifold element approximation. The complex stress intensity factors for bi-material interfacial cracks are numerically evaluated. Good agreement between the numerical results and the analytical solutions for benchmark interfacial crack problems is realized.

Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature

A boundary element formulation is developed to determine the complex stress intensity factors associated with cracks on the interface between dissimilar materials. This represents an extension of the methodology developed previously by the authors for determination of free-edge generalized stress intensity factors on bi-material interfaces, which employs displacements and weighted tractions as primary variables. However, in the present work, the characteristic oscillating stress singularity is addressed through the introduction of complex weighting functions for both displacements and tractions, along with corresponding non-standard numerical quadrature formulas. As a result, this boundary-only approach provides extremely accurate mesh-insensitive solutions for a range of two-dimensional interface crack problems. A number of computational examples are considered to assess the performance of the method in comparison with analytical solutions and previous work on the subject. As a final application, the method is applied to study the scaling behavior of epoxy–metal butt joints.

Evaluation of mixed mode stress intensity factors for interface cracks using EFGM

This paper presents the implementation of element free Galerkin method for the stress analysis of structures having cracks at the interface of two dissimilar materials. The material discontinuity at the interface has been modeled using a jump function with a jump parameter that governs its strength. The jump function enriches the approximation by the addition of special shape function that contains discontinuities in the derivative. The trial and test functions of the weak form are constructed using moving least-square interpolants in each material domain. An intrinsic enrichment criterion with enriched basis has been used to model the crack tip stress fields. The mixed mode (complex) stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The numerical results are obtained for edge and center cracks lying at the bi-material interface, and are found to be in good agreement with the reference solutions for the interfacial crack problems.

Experimental Investigation of Static Interfacial Fracture in Orthotropic Polymer Composite Bimaterials Using Photoelasticity

The effects of static Mode I (opening mode) loading on orthotropic-orthotropic bimaterial interface cracks have been investigated using the experimental technique of transmission photoelasticity. For successful implementation of this experimental technique transparent and birefringent glass fiber reinforced polyester composite materials were developed, enabling the direct observation and recording of photoelastic fringes in the vicinity of the interface crack in the orthotropic bimaterial. It is our belief that this is the first of such experimental investigation. Opening and shearing mode stress-intensity factors and the strain energy release rates have been calculated for various combinations of the bimaterial halves. Results have been verified by mathematically regenerating the observed isochromatic fringe patterns. This study has made it feasible to investigate interfacial fracture in orthotropic-orthotropic bimaterials and orthotropic-isotropic bimaterials using photoelasticity.

Transient dynamic analysis of interface cracks in anisotropic bimaterials by the scaled boundary finite-element method

The transient response of finite bimaterial plates with interface cracks is analyzed directly in the time domain by using the scaled boundary finite-element method. A bimaterial plate is divided into a few subdomains. Only the boundaries of the subdomains are discretized with line elements leading to great flexibility in mesh generation. The displacement and stress fields are expressed as a series solution which separates the singular stress term from other high-order terms. The oscillatory stress singularity in the radial direction emanating from the scaling center is represented analytically. The complex dynamic stress intensity factors are evaluated directly from either the stresses or the crack opening displacements of the singular stress term. Numerical examples of cracked anisotropic bimaterial plates are presented to verify the accuracy of the present technique and to provide additions to the very limited number of reference solutions in the literature.

A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to the mix-mode interface cracks between two dissimilar isotropic materials. The present ES-FEM method uses triangular elements that can be generated automatically for problems even with complicated geometry, and strains are smoothed over the smoothing domains associated with the edges of elements. Considering the stress singularity in the vicinity of the bimaterial interface crack tip is of the inverse square root type together with oscillatory nature, a five-node singular crack tip element is devised within the framework of ES-FEM to construct singular shape functions. Such a singular element can be easily implemented since the derivatives of the singular shape term \({(1/\sqrt r)}\) are not needed. The mix-mode stress intensity factors can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral. The effectiveness of the present singular ES-FEM is demonstrated via benchmark examples for a wide range of material combinations and boundary conditions.

Energy release rates of bi-material interface crack including residual thermal stresses: Application of crack tip element method

Crack tip element method is applied to the formulation of the energy release rate associated with interfacial crack growth of laminates with residual thermal stresses using the Timochenko beam model. Special attention is paid to the energy release rates of double cantilever beam and mixed-mode bending tests of bi-material specimens, and mode-I and mode-II energy release rates are formulated including residual thermal stresses. The derived results are verified by the comparison to finite element analysis, and the effect of residual thermal stresses on the mode mixity of the double cantilever beam and mixed-mode bending tests is discussed.

Transient thermal stress intensity factors of bimaterial interface cracks using refined three-fringe photoelasticity

Transient thermal stresses of a bimaterial specimen with interface edge cracks subjected to heating along an edge is analysed by refined three-fringe photoelasticity (RTFP). Whole-field, noise-free, fringe order estimation using a single colour image is made possible using RTFP combined with colour adaptation. The stress intensity factors (SIFs) of the interface crack are determined through a multiparameter overdeterministic system of equations by a least-squares approach using experimental data collected automatically. The transient SIFs are found to peak to a higher value than in steady state, and the opening mode is found to be dominant. An increase in thermal load causes the crack to propagate, and this is easily visualized on the basis of quantitative fringe order data available for the whole field. The SIFs of a propagating crack are found to be low. The study shows that the crack propagates easily when the opening mode is dominant.

Computation of Thermal Stress Intensity Factors for Bimaterial Interface Cracks Using Domain Integral Method

The concept of domain integral used extensively for J integral has been applied in this work for the formulation of J2 integral for linear elastic bimaterial body containing a crack at the interface and subjected to thermal loading. It is shown that, in the presence of thermal stresses, the Jk domain integral over a closed path, which does not enclose singularities, is a function of temperature and body force. A method is proposed to compute the stress intensity factors for bimaterial interface crack subjected to thermal loading by combining this domain integral with the Jk integral. The proposed method is validated by solving standard problems with known solutions.

Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique

In this study the fracture mechanics parameters, including the strain energy release rate, the stress intensity factors and phase angles, along the curvilinear front of a three-dimensional bimaterial interface crack in electronic packages are considered by using finite element method with the virtual crack closure technique (VCCT). In the numerical procedure normalized complex stress intensity factors and the corresponding phase angles (Rice, J Appl Mech 55:98–103, 1988) are calculated from the crack closure integrals for an opening interface crack tip. Alternative procedures are also described for the cases of crack under inner pressure and crack faces under large-scale contact. Validation for the procedure is performed by comparing numerical results to analytical solutions for the problems of interface crack subjected to either remote tension or mixed loading. The numerical approach is then applied to study interface crack problems in electronic packages. Solutions for semi-circular surface crack and quarter-circular corner crack on the interface of epoxy molding compound and silicon die under uniform temperature excursion are presented. In addition, embedded corner delaminations on the interface of silicon die and underfill in flip-chip package under thermomechanical load are investigated. Based on the distribution of the fracture mechanics parameters along the interface crack front, qualitative predictions on the propensity of interface crack propagation under thermomechanical loads are given.

Influence of Material Characters on Bimaterial Interfacial Crack Energy Release Rates and Stress Intensity Factors

In practical engineering, a structure is often composed of multi-materials, and crack extends on the interface of the two different materials. A calculation model on bimaterial interface crack is developed. Expansion of the crack is described by using the crack energy release rate, and according to analytic method of interfacial crack energy release rates. Simulation is made on the influence of material character, outer loads, plate depth and different material widths beside the interface on the energy release rate and stress intensity factor during the crack expansion processes. Then the relationship curves between crack energy release rate and elastic modulus, outer loads, plate depths of the two materials are plotted respectively, the relationship of stress intensity factor and material widths beside the interface is achieved. And analysis and comparison are made on the interface crack and single material crack.

The multi-domain boundary contour method for interface and dissimilar material problems

A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of 2-D problems does not require any numerical integration at all. The method is thus very computationally effective and accurate as shown in previous related studies. This paper presents a further development of the BCM for multi-domain analysis in 2-D elasticity (multi-domain boundary contour method or MBCM), with potential applications to interface and dissimilar material problems, such as those involving composite materials and bimaterial interface cracks. The primary contribution of this paper is an introduction of a multi-domain technique capable of dealing with a general type of displacement and traction compatibility conditions on the interface(s), including those that arise from structures in which the materials are press-fitted together, and from thermoelasticity in dissimilar materials. Some numerical tests conducted within this work suggest that the proposed multi-domain technique is simple, robust and accurate for bimaterial stress analysis, and also able to provide highly accurate results of both the stress intensity factors K 1 and K 2 for bimaterial interface crack problems.

Thermal weight functions for bi-material interface crack system using energy principles

In this study, Betti’s reciprocal theorem and the principle of superposition are used to obtain weight functions in a two-dimensional bi-material interface crack system for any loading, in general and thermal loading, in particular. It is shown that the general expression of weight functions for bi-materials interface crack problems is of the same type as that found in a homogeneous mixed mode loading case. Furthermore, a computational approach has been developed for calculation of thermal stress intensity factors for bi-material interface cracks subjected to thermal loading under quasi-static uncoupled thermo-elasticity assumption. The thermal weight function (TWF) expression and computational scheme have been validated using three examples given in the available literature.

An Effective Method for Calculation of Corner Stresses With Applications to Plastic IC Packages

Plastic integrated circuit (IC) packages are usually composite components made of multiple bonded materials with different mechanical and thermal properties. High stresses often occur at corners between different components within the packages where discontinuities of geometry or material properties are present. Delaminations usually initiate from these corners when packages undergo adverse thermal or moisture environments. Hence, in order to prevent delaminations from occurring and to improve package reliability performance, it is crucial to accurately and efficiently evaluate the stresses at corners within the package. However, with conventional finite element methods, it is always a challenge to give an accurate description of the stresses at the corners since these corners represent stress singularity points. In this paper, an effective method is developed to precisely evaluate the stresses in the vicinity of internal corners within the packages. A new variable-order singular boundary element is constructed with a built-in accurate description of the stresses at the corner. This method is versatile for solving general corner problems involving wedges, two-material and three-material corners and interfacial cracks that are common in the IC packages. This method is verified by solving a bimaterial interface crack problem with known solution. Comparisons are made with other conventional methods such as displacement-based quarter-point singular elements and normal quadratic elements, on the calculation of interfacial stress intensity factors. Results show that the new method has significant advantages in giving more accurate results with much less computational resources needed. The new method is applied to a typical plastic IC package with multiple internal corners and interface cracks. The possible failure sites and failure modes within the package are predicted and the results agree well with package evaluations done in the industry.

Calculation of transient dynamic stress intensity factors at bimaterial interface cracks using a SBFEM-based frequency-domain approach

A frequency-domain approach based on the semi-analytical scaled boundary finite element method (SBFEM) was developed to calculate dynamic stress intensity factors (DSIFs) at bimaterial interface cracks subjected to transient loading. Because the stress solutions of the SBFEM in the frequency domain are analytical in the radial direction, and the complex stress singularity at the bimaterial interface crack tip is explicitly represented in the stress solutions, the mixed-mode DSIFs were calculated directly by definition. The complex frequency-response functions of DSIFs were then used by the fast Fourier transform (FFT) and the inverse FFT to calculate time histories of DSIFs. A benchmark example was modelled. Good results were obtained by modelling the example with a small number of degrees of freedom due to the semi-analytical nature of the SBFEM.

The use of conservative integral in bi-material interface crack problems subjected to thermal loads

A linear bi-material elastic body containing a crack at the interface and subjected to thermal loading is analyzed. The J 2 line integral, developed for mechanical loads by Khandelwal and Chandra Kishen [Khandelwal, Chandra Kishen, J.M., 2006. Complex variable method of computing J k for bi-material interface cracks. Engineering Fracture Mechanics 73, 1568–1580] is extended to thermal loading. This method, used in conjunction with the finite element method, is shown to be useful in the prediction of stress intensity factors for cracks lying at the interface of two dissimilar materials and subjected to any type of thermal loading.

Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method. The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r −1/2±i ε or the non-oscillating singularity r −1/2± κ . Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.